Witt vector#Motivation

In the 19th century, Ernst Eduard Kummer studied cyclic extensions of fields as part of his work on Fermat's Last Theorem. This led to the subject known as Kummer theory. Let $k$ be a field containing a primitive $n$-th root of unity. Kummer theory classifies degree $n$ cyclic field extensions $K$ of $k$. Such fields are in bijection with order $n$ cyclic groups $\backslash Delta\; \backslash subseteq\; k^\backslash times\; /\; (k^\backslash times)^n$, where $\backslash Delta$ corresponds to $K=k(\backslash sqrt[n]\{\backslash Delta\}\backslash ,)$.

But suppose that $k$ has characteristic $p$. The problem of studying degree $p$ extensions of $k$, or more generally degree $p^n$ extensions, may appear superficially similar to Kummer theory. However, in this situation, $k$ cannot contain a primitive $p$-th root of unity. If $x$ is a $p$-th root of unity in $k$, then it satisfies $x^p=1$. But consider the expression $(x-1)^p=0$. By expanding using binomial coefficients, the operation of raising to the $p$-th power, known here as the Frobenius homomorphism, introduces the factor $p$ to every coefficient except the first and the last, and so modulo $p$ these equations are the same. Therefore $x\; =\; 1$. Consequently, Kummer theory is never applicable to extensions whose degree is divisible by the characteristic.

The case where the characteristic divides the degree is today called Artin–Schreier theory because the first progress was made by Artin and Schreier. Their initial motivation was the Artin–Schreier theorem, which characterizes the real closed fields as those whose absolute Galois group has order two.Artin, Emil and Schreier, Otto, Über eine Kennzeichnung der reell abgeschlossenen Körper, Abh. Math. Sem. Hamburg 3 (1924). This inspired them to ask what other fields had finite absolute Galois groups. In the midst of proving that no other such fields exist, they proved that degree $p$ extensions of a field $k$ of characteristic $p$ were the same as splitting fields of Artin–Schreier polynomials. These are by definition of the form $x^p-x-a.$ By repeating their construction, they described degree $p^2$ extensions. Abraham Adrian Albert used this idea to describe degree $p^n$ extensions. Each repetition entailed complicated algebraic conditions to ensure that the field extension was normal.A. A. Albert, Cyclic fields of degree $p^n$ over $F$ of characteristic $p$, Bull. Amer. Math. Soc. 40 (1934).

SchmidSchmid, H. L., Zyklische algebraische Funktionenkörper vom Grad pⁿ über endlichen Konstantenkörper der Charakteristik p, Crelle 175 (1936). generalized further to non-commutative cyclic algebras of degree $p^n$. In the process of doing so, certain polynomials related to the addition of $p$-adic integers appeared. Witt seized on these polynomials. By using them systematically, he was able to give simple and unified constructions of degree $p^n$ field extensions and cyclic algebras. Specifically, he introduced a ring now called $W\_n(k)$, the ring of $n$-truncated $p$-typical Witt vectors. This ring has $k$ as a quotient, and it comes with an operator $F$ which is called the Frobenius operator since it reduces to the Frobenius operator on $k$. Witt observed that the degree $p^n$ analog of Artin–Schreier polynomials is

:$F(x)-x-a$,

where $a\; \backslash in\; W\_n(k)$. To complete the analogy with Kummer theory, define $\backslash wp$ to be the operator $x\; \backslash mapsto\; F(x)-x.$ Then the degree $p^n$ extensions of $k$ are in bijective correspondence with cyclic subgroups $\backslash Delta\; \backslash subseteq\; W\_n(k)\; /\; \backslash wp(W\_n(k))$ of order $p^n$, where $\backslash Delta$ corresponds to the field $k(\backslash wp^\{-1\}(\backslash Delta))$.

Any $p$-adic integer (an element of $\backslash Z\_p$, not to be confused with $\backslash Z/p\backslash Z=\backslash mathbb\{F\}\_p$) can be written as a power series $a\_0+a\_1p^1+a\_2p^2+\backslash cdots$, where the $a\_i$ are usually taken from the integer interval $[0,p-1]=\backslash \{0,1,2,\backslash ldots,p-1\backslash \}$. It can be difficult to provide an algebraic expression for addition and multiplication using this representation, as one faces the problem of carrying between digits. However, taking representative coefficients $a\_i\backslash in\; [0,p-1]$ is only one of many choices, and Hensel himself (the creator of $p$-adic numbers) suggested the roots of unity in the field as representatives. These representatives are therefore the number $0$ together with the $(p-1)^\{\backslash text\{st\}\}$ roots of unity; that is, the solutions of $x^p\; -\; x\; =\; 0$ in $\backslash Z\_p$, so that $a\_i=a\_i^p$. This choice extends naturally to ring extensions of $\backslash Z\_p$ in which the residue field is enlarged to $\backslash mathbb\{F\}\_q$ with $q=p^f$, some power of $p$. Indeed, it is these fields (the fields of fractions of the rings) that motivated Hensel's choice. Now the representatives are the $q$ solutions in the field to $x^q-x=0$. Call the field $\backslash Z\_p(\backslash eta)$, with $\backslash eta$ an appropriate primitive $(q-1)^\{\backslash text\{th\}\}$ root of unity (over $\backslash Z\_p$). The representatives are then $0$ and $\backslash eta^i$ for $0\backslash leq\; i\backslash leq\; q-2$. Since these representatives form a multiplicative set they can be thought of as characters. Some thirty years after Hensel's works, Teichmüller studied these characters, which now bear his name, and this led him to a characterisation of the structure of the whole field in terms of the residue field. These Teichmüller representatives can be identified with the elements of the finite field $\backslash mathbb\{F\}\_q$ of order $q$ by taking residues modulo $p$ in $\backslash Z\_p(\backslash eta)$, and elements of $\backslash mathbb\{F\}\_q^\backslash times$ are taken to their representatives by the Teichmüller character $\backslash omega:\backslash mathbb\{F\}\_q^\backslash times\backslash to\backslash Z\_p(\backslash eta)^\backslash times$. This operation identifies the set of integers in $\backslash Z\_p(\backslash eta)$ with infinite sequences of elements of $\backslash omega(\backslash mathbb\{F\}\_q^\backslash times)\backslash cup\backslash \{0\backslash \}$.

Taking those representatives, the expressions for addition and multiplication can be written in closed form. The following problem (stated for the simplest case: $q=p$): given two infinite sequences of elements of $\backslash omega(\backslash mathbb\{F\}\_p^\backslash times)\backslash cup\backslash \{0\backslash \}$, describe their sum and product as p-adic integers explicitly. This problem was solved by Witt using Witt vectors.{{Citation needed|date=January 2025}}

The ring of $p$-adic integers $\backslash Z\_p$ is derived from the finite field $\backslash mathbb\{F\}\_p=\backslash Z\; /p\backslash Z$ using a construction which naturally generalizes to the Witt vector construction.

The ring $\backslash Z\_p$ of p-adic integers can be understood as the inverse limit of the rings $\backslash Z\; /p^i\backslash Z$ taken along the projections. Specifically, it consists of the sequences $(n\_0,n\_1,\backslash ldots)$ with $n\_i\backslash in\backslash Z/p^\{i+1\}\backslash Z$, such that $n\_j\backslash equiv\; n\_i\backslash bmod\; p^\{i+1\}$ for $j\backslash ge\; i$. That is, each successive element of the sequence is equal to the previous elements modulo a lower power of p; this is the inverse limit of the projections $\backslash Z/p^\{i+1\}\backslash Z\backslash to\backslash Z/p^i\backslash Z$.

The elements of $\backslash Z\_p$ can be expanded as (formal) power series in $p$

:$a\_0+a\_1p^1+a\_2p^2+\backslash cdots$,

where the coefficients $a\_i$ are taken from the integer interval $[0,p-1]=\backslash \{0,1,\backslash ldots,p-1\backslash \}$. This power series usually will not converge in $\backslash R$ using the standard metric on the reals, but it will converge in $\backslash Z\_p$ with the p-adic metric.

Letting $a+b$ be denoted by $c$, the following definition can be considered for addition:

:$\backslash begin\{align\}$

c_0 &\equiv a_0+b_0 && \bmod p \\

c_0+c_1 p &\equiv (a_0+b_0) + (a_1 +b_1)p && \bmod p^2 \\

c_0+c_1 p+c_2 p^2 &\equiv (a_0+b_0) + (a_1 +b_1)p+(a_2+b_2) p^2 &&\bmod p^3

\end{align}

and a similar definition for multiplication can be made. However, this is not a closed formula, since the new coefficients are not in the allowed set $[0,p-1]$.

There is a coefficient subset of $\backslash Z\_p$ which does yield closed formulas, the Teichmüller representatives: zero together with the $(p-1)^\{\backslash text\{th\}\}$ roots of unity. They can be explicitly calculated (in terms of the original coefficient representatives $[0,p-1]$) as roots of $x^\{p-1\}-1=0$ through Hensel lifting, the p-adic version of Newton's method. For example, in $\backslash Z\_5$, to calculate the representative of 2, one starts by finding the unique solution of $x^4-1=0$ in $\backslash Z/25\backslash Z$ with $x\backslash equiv2\backslash bmod5$; one gets 7. Repeating this in $\backslash Z/125\backslash Z$, with the conditions $x^4-1=0$ and $x\backslash equiv7\backslash bmod25$, gives 57, and so on; the resulting Teichmüller representative of 2, denoted $\backslash omega(2)$, is the sequence

$\backslash omega(2)=(2,7,57,\backslash ldots)\backslash in\; W(\backslash mathbb\{F\}\_5)$.

This algorithm shows that for every $j\backslash in[0,p-1]$, there is one Teichmüller representative with $a\_0=j$, which is denoted $\backslash omega(j)$. This defines the Teichmüller character $\backslash omega:\backslash mathbb\{F\}\_p^*\backslash to\backslash Z\_p^*$ as a (multiplicative) group homomorphism, which moreover satisfies $m\backslash circ\; \backslash omega=\backslash mathrm\{id\}\_\{\backslash mathbb\{F\}\_p\}$ if one lets $m:\backslash Z\_p\backslash to\; \backslash Z\_p/p\backslash Z\_p\; \backslash cong\; \backslash mathbb\{F\}\_p$ denote the canonical projection. Note however that $\backslash omega$ is not additive, as the sum need not be a representative. Despite this, if $\backslash omega(k)\; \backslash equiv\; \backslash omega(i)+\backslash omega(j)\backslash bmod\; p$ in $\backslash Z\_p,$ then $i+j=k$ in $\backslash mathbb\{F\}\_p.$

Because of this one-to-one correspondence given by $\backslash omega$, one can expand every p-adic integer as a power series in p with coefficients taken from the Teichmüller representatives. An explicit algorithm can be given, as follows. Write the Teichmüller representative as $\backslash omega(t\_0)=t\_0+t\_1p^1+t\_2p^2+\backslash cdots$. Then, if one has some arbitrary p-adic integer of the form $x=x\_0+x\_1p^1+x\_2p^2+\backslash cdots$, one takes the difference $x-\backslash omega(x\_0)=x\text{'}\_1p^1+x\text{'}\_2p^2+\backslash cdots$, leaving a value divisible by $p$. Hence, $x-\backslash omega(x\_0)=0\backslash bmod\; p$. The process is then repeated, subtracting $\backslash omega(x\text{'}\_1)p$ and proceed likewise. This yields a sequence of congruences

:

so that

:$x\backslash equiv\backslash sum\_\{j=0\}^i\backslash omega(\backslash bar\{x\}\_j)p^j\backslash bmod\; p^\{i+1\}$

and $i\text{'}\; >\; i$ implies

:$\backslash sum\_\{j\; =\; 0\}^\{i\text{'}\}\; \backslash omega(\backslash bar\{x\}\_j)p^j\; \backslash equiv\; \backslash sum\_\{j=0\}^i\; \backslash omega(\backslash bar\{x\}\_j)p^j\; \backslash bmod\; p^\{i+1\}$

for

:$\backslash bar\{x\}\_i:=m\backslash left(\backslash frac\{x-\backslash sum\_\{j=0\}^\{i-1\}\backslash omega(\backslash bar\{x\}\_j)p^j\}\{p^i\}\backslash right)$.

This obtains a power series for each residue of $x$ modulo powers of $p$, but with coefficients in the Teichmüller representatives rather than $\backslash \{0,\backslash ldots,p-1\backslash \}$.

:$\backslash sum\_\{j=0\}^\backslash infty\backslash omega(\backslash bar\{x\}\_j)p^j=x$,

since

:$p^\{i+1\}\backslash mid\; x-\backslash sum\_\{j=0\}^i\backslash omega(\backslash bar\{x\}\_j)p^j$

for all $i$ as $i\backslash to\backslash infty$, so the difference tends to 0 with respect to the p-adic metric. The resulting coefficients will typically differ from the $a\_i$ modulo $p^i$ except the first one.

The Teichmüller coefficients have the key additional property that $\backslash omega(\backslash bar\{x\}\_i)^p=\backslash omega(\backslash bar\{x\}\_i),$ which is missing for the numbers in $[0,p-1]$. This can be used to describe addition, as follows. Consider the equation $c\; =\; a+b$ in $\backslash mathbb\{Z\}\_p$ and let the coefficients $a\_i,\; b\_i,\; c\_i\; \backslash in\; \backslash mathbb\{Z\}\_p$ now be as in the Teichmüller expansion. Since the Teichmüller character is not additive, $c\_0=a\_0+b\_0$ is not true in $\backslash Z\_p$, but it holds in $\backslash mathbb\{F\}\_p$, as the first congruence implies. In particular,

:$c\_0^p\backslash equiv(a\_0+b\_0)^p\backslash bmod\; p^2$

and thus

:$c\_0-a\_0-b\_0\backslash equiv(a\_0+b\_0)^p-a\_0-b\_0\backslash equiv\backslash binom\{p\}\{1\}a\_0^\{p-1\}b\_0+\backslash cdots+\backslash binom\{p\}\{p-1\}a\_0b\_0^\{p-1\}\backslash bmod\; p^2$.

Since the binomial coefficient $\backslash binom\{p\}\{i\}$ is divisible by $p$, this gives

:$c\_1\backslash equiv\; a\_1+b\_1-a\_0^\{p-1\}b\_0-\backslash frac\{p-1\}\{2\}a\_0^\{p-2\}b\_0^2-\backslash cdots-a\_0b\_0^\{p-1\}\backslash bmod\; p$.

This completely determines $c\_1$ by the lift. Moreover, the congruence modulo $p$ indicates that the calculation can actually be done in $\backslash mathbb\{F\}\_p,$ satisfying the basic aim of defining a simple additive structure.

For $c\_2$ this step can be cumbersome. Write

:$c\_1=c\_1^p\backslash equiv\backslash left(a\_1+b\_1-a\_0^\{p-1\}b\_0-\backslash frac\{p-1\}\{2\}a\_0^\{p-2\}b\_0^2-\backslash cdots-a\_0b\_0^\{p-1\}\backslash right)^p\backslash bmod\; p^2$.

Just as for $c\_0$, a single $p$th power is not enough: one must take

:$c\_0=c\_0^\{p^2\}\backslash equiv(a\_0+b\_0)^\{p^2\}\; \backslash bmod\; p^3.$

However, $\backslash binom\{p^2\}\{i\}$ is not in general divisible by $p^2$, but it is divisible when $i=pd$, in which case $a^ib^\{p^2-i\}=a^db^\{p-d\}$ combined with similar monomials in $c\_1^p$ will make a multiple of $p^2$.

At this step, one works with addition of the form

:$\backslash begin\{align\}$

c_0 &\equiv a_0+b_0 && \bmod p \\

c_0^p+c_1 p &\equiv a_0^p+a_1 p+b_0^p+b_1 p && \bmod p^2 \\

c_0^{p^2}+c_1^p p+c_2 p^2 &\equiv a_0^{p^2}+a_1^p p+a_2 p^2+b_0^{p^2}+b_1^p p+b_2 p^2 && \bmod p^3

\end{align}

This motivates the definition of Witt vectors.

Fix a prime number p. A Witt vector{{Cite journal|last=Illusie|first=Luc|date=1979|title=Complexe de de Rham-Witt et cohomologie cristalline|url=http://www.numdam.org/item/ASENS_1979_4_12_4_501_0/|journal=Annales scientifiques de l'École Normale Supérieure|language=fr|volume=12|issue=4|pages=501–661|doi=10.24033/asens.1374|doi-access=free}} over a commutative ring $R$ (relative to the prime $p$) is a sequence $(X\_0,X\_1,X\_2,\backslash ldots)$ of elements of $R$. The Witt polynomials $W\_i$ can be defined by

1. $W\_0=X\_0$
2. $W\_1=X\_0^p+pX\_1$
3. $W\_2=X\_0^\{p^2\}+pX\_1^p+p^2X\_2$

and in general

:$W\_n=\backslash sum\_\{i=0\}^np^iX\_i^\{p^\{n-i\}\}$.

The $W\_n$ are called the ghost components of the Witt vector $(X\_0,X\_1,X\_2,\backslash ldots)$, and are usually denoted by $X^\{(n)\}$; taken together, the $W\_n$ define the ghost map to $\backslash prod\_\{i=0\}^\backslash infty\; R$. If $R$ is p-torsionfree, then the ghost map is injective and the ghost components can be thought of as an alternative coordinate system for the $R$-module of sequences (though note that the ghost map is not surjective unless $R$ is p-divisible).

The ring of (p-typical) Witt vectors $W(R)$ is defined by componentwise addition and multiplication of the ghost components. That is, that there is a unique way to make the set of Witt vectors over any commutative ring $R$ into a ring such that:

1. the sum and product are given by polynomials with integer coefficients that do not depend on $R$, and
2. projection to each ghost component is a ring homomorphism from the Witt vectors over $R$, to $R$.

In other words,

• $(X+Y)\_i$ and $(XY)\_i$ are given by polynomials with integer coefficients that do not depend on R, and

• $X^\{(i)\}+Y^\{(i)\}\; =\; (X+Y)^\{(i)\}$ and $X^\{(i)\}Y^\{(i)\}\; =\; (XY)^\{(i)\}.$

The first few polynomials giving the sum and product of Witt vectors can be written down explicitly. For example,

:$(X\_0,X\_1,\backslash ldots)+(Y\_0,Y\_1,\backslash ldots)=(X\_0+Y\_0,\; X\_1+Y\_1-((X\_0+Y\_0)^p\; -\; X\_0^p\; -Y\_0^p)/p,\backslash ldots)$

:$(X\_0,X\_1,\backslash ldots)\backslash times(Y\_0,Y\_1,\backslash ldots)=(X\_0\; Y\_0,\; X\_0^p\; Y\_1+X\_1\; Y\_0^p+p\; X\_1\; Y\_1,\backslash ldots)$

These are to be understood as shortcuts for the actual formulas: if for example the ring $R$ has characteristic $p$, the division by $p$ in the first formula above, the one by $p^2$ that would appear in the next component and so forth, do not make sense. However, if the $p$-power of the sum is developed, the terms $X\_0^p+Y\_0^p$ are cancelled with the previous ones and the remaining ones are simplified by $p$, no division by $p$ remains and the formula makes sense. The same consideration applies to the ensuing components.

As would be expected, the identity element in the ring of Witt vectors $W(A)$ is the element

$\backslash underline\{1\}\; =\; (1,0,0,\backslash ldots)$

$\backslash underline\{1\}\; +\; \backslash underline\{1\}\; =\; (2,4,\backslash ldots)$

$\backslash begin\{align\}$

2 &= 1 + 1\\

4 &= -\frac{32 - 1 - 1}{5} \mod 5 \\

&\cdots

\end{align}

$\backslash underline\{x\}a\; =\; (xa\_0,x^pa\_1,\backslash ldots,x^\{p^n\}a\_n,\backslash ldots)$

• The Witt ring of any commutative ring $R$ in which $p$ is invertible, is isomorphic to $R^\backslash N$ (the product of a countable number of copies of $R$). The Witt polynomials always give a homomorphism from the ring of Witt vectors to $R^\backslash N$, and if $p$ is invertible this homomorphism is an isomorphism.

• The Witt ring $W(\backslash mathbb\{F\}\_p)\; \backslash cong\; \backslash mathbb\{Z\}\_p$ of the finite field of order $p$ is the ring of $p$-adic integers written in terms of the Teichmüller representatives, as demonstrated above.

• The Witt ring $W(\backslash mathbb\{F\}\_q)\; \backslash cong\; \backslash mathcal\{O\}\_K$ of a finite field of order $p^n$ is the ring of integers of the unique unramified extension of degree $n$ of the ring of $p$-adic numbers $K/\backslash mathbb\{Q\}\_p$. Note $K\; \backslash cong\; \backslash mathbb\{Q\}\_p(\backslash mu\_\{q-1\})$ for $\backslash mu\_\{q-1\}$ the $(q-1)$-st root of unity, hence $W(\backslash mathbb\{F\}\_q)\; \backslash cong\; \backslash mathbb\{Z\}\_p[\backslash mu\_\{q-1\}]$.

• The truncated Witt ring $W\_\{n+1\}(\backslash mathbb\{F\}\_p[t])$ can be described as{{Cite journal|last=Illusie|first=Luc|date=1979|title=Complexe de de Rham-Witt et cohomologie cristalline|location=Section I.2.B|url=http://www.numdam.org/item/ASENS_1979_4_12_4_501_0/|journal=Annales scientifiques de l'École Normale Supérieure|language=fr|volume=12|issue=4|pages=501–661|doi=10.24033/asens.1374|doi-access=free}}

$$

\begin{array}{rcl}

W_{n+1}(\mathbb{F}_p[t])

&\cong&

\left \{

\sum a_i t^{i/p^n}

\in

(\mathbb{Z}/p^{n+1})[t^{1/p^n}]

\ :\

p i a_i = 0

\

\textrm{ mod }

\

p^{n+1}

\right \} \\

&=&

(\mathbb{Z}/p^n)[t] + p(\mathbb{Z}/p^n)[t^{1/p}] + p^2(\mathbb{Z}/p^n)[t^{1/p^2}] + \dots + p^n(\mathbb{Z}/p^n)[t^{1/p^n}]

\end{array}

The Witt vectors are the inverse limit along the canonical projections

$W(\backslash mathbb\{F\}\_p[t])\; =\; \backslash lim\; (\backslash dots\; \backslash to\; W\_\{n+1\}(\backslash mathbb\{F\}\_p[t])\; \backslash to\; W\_\{n\}(\backslash mathbb\{F\}\_p[t])\; \backslash to\; \backslash dots\; \backslash to\; W\_1(\backslash mathbb\{F\}\_p[t])\; =\; \backslash mathbb\{F\}\_p[t]).$

Here the transition homomorphisms are induced by reduction $(\backslash mathbb\{Z\}/p^\{n+1\})[t^\{1/p^n\}]\; \backslash to\; (\backslash mathbb\{Z\}/p^\{n\})[t^\{1/p^n\}]$.

The Witt polynomials for different primes $p$ are special cases of universal Witt polynomials, which can be used to form a universal Witt ring (not depending on a choice of prime $p$). Define the universal Witt polynomials $W\_n$ for $n\backslash geq1$ by

1. $W\_1=X\_1$
2. $W\_2=X\_1^2+2X\_2$
3. $W\_3=X\_1^3+3X\_3$
4. $W\_4=X\_1^\{4\}+2X\_2^2+4X\_4$

and in general

:$W\_n=\backslash sum\_\{d|n\}dX\_d^\{n/d\}$.

Again, $(W\_1,W\_2,W\_3,\backslash ldots)$ is called the vector of ghost components of the Witt vector $(X\_1,X\_2,X\_3,\backslash ldots)$, and is usually denoted by $(X^\{(1)\},X^\{(2)\},X^\{(3)\},\backslash ldots)$.

These polynomials can be used to define the ring of universal Witt vectors or big Witt ring of any commutative ring $R$ in much the same way as above (so the universal Witt polynomials are all homomorphisms to the ring $R$).

Witt also provided another approach using generating functions.{{cite book |last=Lang |first=Serge |title=Algebra |url=https://archive.org/details/undergraduatealg00lang_077 |url-access=limited |publisher=Springer|edition = 3rd |date=September 19, 2005 |pages=[https://archive.org/details/undergraduatealg00lang_077/page/n340 330] |chapter=Chapter VI: Galois Theory |isbn=978-0-387-95385-4}}

Let $X$ be a Witt vector and define

:$f\_X(t)=\backslash prod\_\{n\backslash ge\; 1\}(1-X\_n\; t^n)=\backslash sum\_\{n\backslash ge\; 0\}A\_n\; t^n$

For $n\backslash ge\; 1$ let $\backslash mathcal\{I\}\_n$ denote the collection of subsets of $\backslash \{1,2,\backslash ldots,n\backslash \}$ whose elements add up to $n$. Then

:$A\_n=\backslash sum\_\{I\backslash in\backslash mathcal\{I\}\_n\}(-1)^$

One can get the ghost components by taking the logarithmic derivative:

:$\backslash begin\{align\}$

-t\frac{d}{dt}\log f_X(t)&= -t\frac{d}{dt} \sum_{n\ge 1} \log(1-X_n t^n) \\

&=t \frac{d}{dt}\sum_{n\ge 1}\sum_{d\ge 1}\frac{X_n^d t^{nd}}{d}\\

&=\sum_{n\ge 1}\sum_{d\ge 1} n X_n^d t^{nd} \\

&=\sum_{m\ge 1}\sum_{d|m}dX_{d}^{m/d}t^m \\

&=\sum_{m\ge 1}X^{(m)}t^m

\end{align}

Now one can see $f\_\{Z\}(t)=f\_X(t)\; f\_Y(t)$ if $Z=X+Y$. So that

:$C\_n=\backslash sum\_\{0\backslash le\; i\backslash le\; n\}A\_n\; B\_\{n-i\},$

if $A\_n,B\_n,C\_n$ are the respective coefficients in the power series $f\_X(t),f\_Y(t),f\_Z(t)$. Then

:$Z\_n=\backslash sum\_\{0\backslash le\; i\backslash le\; n\}A\_n\; B\_\{n-i\}-\backslash sum\_\{I\backslash in\backslash mathcal\{I\}\_n,I\backslash ne\backslash \{n\backslash \}\}(-1)^$

Since $A\_n$ is a polynomial in $X\_1,\; \backslash ldots,\; X\_n$ and likewise for $B\_n$, one can show by induction that $Z\_n$ is a polynomial in $X\_1,\; \backslash ldots,\; X\_n,\; Y\_1,\; \backslash ldots,\; Y\_n.$

If $W=XY$ is set, then

:$-t\backslash frac\{d\}\{dt\}\backslash log\; f\_W(t)=-\backslash sum\_\{m\backslash ge\; 1\}X^\{(m)\}Y^\{(m)\}t^m$.

but

:$\backslash sum\_\{m\backslash ge\; 1\}X^\{(m)\}Y^\{(m)\}t^m=\backslash sum\_\{m\backslash ge\; 1\}\backslash sum\_\{d|m\}d\; X\_d^\{m/d\}\backslash sum\_\{e|m\}e\; Y\_e^\{m/e\}t^m$.

Now 3-tuples $\{m,d,e\}$ with $m\; \backslash in\; \backslash Z^+,\; d\; \backslash ,|\backslash ,\; m,\; e\; \backslash ,|\backslash ,\; m$ are in bijection with 3-tuples $\{d,e,n\}$ with $d,e,n\; \backslash in\; \backslash Z^+$, via $n\; =\; m/[d,e]$ ($[d,e]$ is the least common multiple), the series becomes

:$\backslash sum\_\{d,e\backslash ge\; 1\}d\; e\backslash sum\_\{n\backslash ge\; 1\}\; \backslash left\; (X\_d^\{\backslash frac\{[d,e]\}\{d\}\}\; Y\_e^\{\backslash frac\{[d,e]\}\{e\}\}\; t^\{[d,e]\}\; \backslash right\; )^n=-t\backslash frac\{d\}\{dt\}\backslash log\backslash prod\_\{d,e\backslash ge\; 1\}\; \backslash left\; (1-X\_d^\{\backslash frac\{[d,e]\}\{d\}\}Y\_e^\{\backslash frac\{[d,e]\}\{e\}\}\; t^\{[d,e]\}\; \backslash right\; )^\{\backslash frac\{de\}\{[d,e]\}\}$

so that

:$f\_W(t)=\backslash prod\_\{d,e\backslash ge\; 1\}\; \backslash left\; (1-X\_d^\{\backslash frac\{[d,e]\}\{d\}\}Y\_e^\{\backslash frac\{[d,e]\}\{e\}\}\; t^\{[d,e]\}\; \backslash right\; )^\{\backslash frac\{de\}\{[d,e]\}\}=\backslash sum\_\{n\backslash ge\; 0\}D\_n\; t^n$,

where $D\_n$ are polynomials of $X\_1,\; \backslash ldots,\; X\_n,\; Y\_1,\; \backslash ldots,\; Y\_n.$ So by similar induction,

:$f\_W(t)=\backslash prod\_\{n\backslash ge\; 1\}(1-W\_n\; t^n),$

then $W\_n$ can be solved as polynomials of $X\_1,\backslash ldots,\; X\_n,\; Y\_1,\backslash ldots,\; Y\_n$.

The map taking a commutative ring $R$ to the ring of Witt vectors over $R$ (for a fixed prime $p$) is a functor from commutative rings to commutative rings, and is also representable, so it can be thought of as a ring scheme, called the Witt scheme, over $\backslash operatorname\{Spec\}(\backslash Z).$ The Witt scheme can be canonically identified with the spectrum of the ring of symmetric functions.

Similarly, the rings of truncated Witt vectors, and the rings of universal Witt vectors correspond to ring schemes, called the truncated Witt schemes and the universal Witt scheme.

Moreover, the functor taking the commutative ring $R$ to the set $R^n$ is represented by the affine space $\backslash mathbb\{A\}\_\{\backslash Z\}^n$, and the ring structure on $R^n$ makes $\backslash mathbb\{A\}\_\{\backslash Z\}^n$ into a ring scheme denoted $\backslash underline\{\backslash mathcal\{O\}\}^n$. From the construction of truncated Witt vectors, it follows that their associated ring scheme $\backslash mathbb\{W\}\_n$ is the scheme $\backslash mathbb\{A\}\_\{\backslash Z\}^n$ with the unique ring structure such that the morphism $\backslash mathbb\{W\}\_n\backslash to\; \backslash underline\{\backslash mathcal\{O\}\}^n$ given by the Witt polynomials is a morphism of ring schemes.

Over an algebraically closed field of characteristic 0, any unipotent abelian connected algebraic group is isomorphic to a product of copies of the additive group $G\_a$. The analogue of this for fields of characteristic $p$ is false: the truncated Witt schemes are counterexamples. (They are made into algebraic groups by using the additive structure instead of multiplication.) However, these are essentially the only counterexamples: over an algebraically closed field of characteristic $p$, any unipotent abelian connected algebraic group is isogenous to a product of truncated Witt group schemes.

André Joyal explicated the universal property of the (p-typical) Witt vectors.{{Cite journal |last=Joyal |first=André |date=1985 |title=δ-anneaux et vecteurs de Witt |journal=C.R. Math. Rep. Acad. Sci. Canada |volume=7 |issue=3 |pages=177-182}} The basic intuition is that the formation of Witt vectors is the universal way to deform a characteristic p ring to characteristic 0 together with a lift of its Frobenius endomorphism.{{Cite web |title=Is there a universal property for Witt vectors? |url=https://mathoverflow.net/questions/58/is-there-a-universal-property-for-witt-vectors |access-date=2022-09-06 |website=MathOverflow |language=en}} To make this precise, define a $\backslash delta$-ring $(R,\; \backslash delta)$ to consist of a commutative ring $R$ together with a map of sets $\backslash delta:\; R\; \backslash to\; R$ that is a p-derivation, so that $\backslash delta$ satisfies the relations

• $\backslash delta(0)\; =\; \backslash delta(1)\; =\; 0$;
• $\backslash delta(x\; y)\; =\; x^p\; \backslash delta(y)\; +\; y^p\; \backslash delta(x)\; +\; p\; \backslash delta(x)\; \backslash delta(y)$;
• $\backslash delta(x\; +\; y)\; =\; \backslash delta(x)\; +\; \backslash delta(y)\; +\; \backslash frac\{x^p+y^p-(x+y)^p\}\{p\}$.

The definition is such that given a $\backslash delta$-ring $(R,\; \backslash delta)$, if one defines the map $\backslash phi:\; R\; \backslash to\; R$ by the formula $\backslash phi(x)\; =\; x^p\; +\; p\; \backslash delta(x)$, then $\backslash phi$ is a ring homomorphism lifting Frobenius on $R/p$. Conversely, if $R$ is p-torsionfree, then this formula uniquely defines the structure of a $\backslash delta$-ring on $R$ from that of a Frobenius lift. One may thus regard the notion of $\backslash delta$-ring as a suitable replacement for a Frobenius lift in the non-p-torsionfree case.

The collection of $\backslash delta$-rings and ring homomorphisms thereof respecting the $\backslash delta$-structure assembles to a category $\backslash mathrm\{CRing\}\_\{\backslash delta\}$. One then has a forgetful functor$$U:\; \backslash mathrm\{CRing\}\_\{\backslash delta\}\; \backslash to\; \backslash mathrm\{CRing\}$$whose right adjoint identifies with the functor $W$ of Witt vectors. The functor $U$ creates limits and colimits and admits an explicitly describable left adjoint as a type of free functor; from this, it can be shown that $\backslash mathrm\{CRing\}\_\{\backslash delta\}$ inherits local presentability from $\backslash mathrm\{CRing\}$ so that one can construct the functor $W$ by appealing to the adjoint functor theorem.

One further has that $W$ restricts to a fully faithful functor on the full subcategory of perfect rings of characteristic p. Its image then consists of those $\backslash delta$-rings that are perfect (in the sense that the associated map $\backslash phi$ is an isomorphism) and whose underlying ring is p-adically complete.{{Cite web |last=Bhatt |first=Bhargav |date=October 8, 2018 |title=Lecture II: Delta rings |url=https://www.math.ias.edu/~bhatt/teaching/prismatic-columbia/lecture2-delta-rings.pdf |url-status=live |archive-url=https://web.archive.org/web/20220906153816/https://www.math.ias.edu/~bhatt/teaching/prismatic-columbia/lecture2-delta-rings.pdf |archive-date=September 6, 2022}}

• p-derivation
• Formal group
• Artin–Hasse exponential
• Necklace ring

{{reflist}}

• [https://web.archive.org/web/20190112130142/http://www.claymath.org/sites/default/files/brinon_witt.pdf Notes on Witt vectors: a motivated approach] - Basic notes giving the main ideas and intuition. Best to start here!
• The Theory of Witt Vectors - Elementary introduction to the theory.
• [http://www.numdam.org/item/ASENS_1979_4_12_4_501_0/ Complexe de de Rham-Witt et cohomologie cristalline] - Note he uses a different but equivalent convention as in this article. Also, the main points in the introduction are still valid.

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Category:Ring theory

Category:Algebraic groups

Category:Combinatorics on words